Fock Spaces for the q-Dunkl Kernel

Fethi Soltani

doi:10.4236/apm.2012.23023

Abstract

In this work, we introduce a class of Hilbert spaces of entire functions on the disk , 0＜q＜1 , with reproducing kernel given by the q-Dunkl kernel . The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Q by z and the q-Dunkl operator on the Fock space ; and we prove that these operators are adjoint-operators and continuous from this space into itself.

Highlights

Fock spaceis the Hilbert space of entire functions f z n 0 an z n on such that f : an 2 n! . n 0This space was introduced by Bargmann in [2] and it was the aim of many works [1]
We introduce a class of Hilbert spaces of entire functions on the disk
We study the multiplication operator Q by z and the q-Dunkl operator q, on the Fock space q, ; and we prove that these operators are adjoint-operators and continuous from this space into itself

Summary

Introduction

1 2, and the multiplication by z are densely defined, closed and adjoint-operators. Cuss some properties of a class of Fock spaces associated to the q-Dunkl kernel and we give some applications. The q-Fock space q, has a reproducing kernel q, given by q, w, z E wz; q2 ; w, z. Using this property, we prove that the space q, is a Hilbert space and we give an Hilbert basis. Using the previous results, we consider the multiplication operator Q by z and the q-Dunkl operator q, on the Fock space q, , and we prove that these. The Dunkl kernel E x; q2 can be expanded in a power series in the form (1)

This proves that

Thus we obtain